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GE 331-Lecture 17

# GE 331-Lecture 17 - Bivariate Normal Distribution Recall...

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IE 300/GE 331 Lecture 17 Negar Kiyavash, UIUC 1 Recall: Normal distribution: Let R.V.s U and V be independent normal, what can we say about X=aU+bV and Y=cU+dV? (a, b, c, d are scalars) They are Normal too, but not independent. In fact any linear combination of X and Y is normal! Bivariate Normal Distribution = 2 2 2 ) ( exp 2 1 ) ( σ μ σ π x x f

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IE 300/GE 331 Lecture 17 Negar Kiyavash, UIUC 2 Bivariate Normal Distribution U and V are independent normal, X=aU+bV and Y=cU+dV (a, b, c, d are scalars) X,Y are jointly normal (Gaussian) (Attention jointly normal is not same as normal!) The Joint PDF is called a bivariate normal distribution: with = ) ( ) ( 2 1 exp | | 2 1 ) ( 1 T μ x Σ μ x Σ x π f = = = 2 2 , , y y x y x x Y X y x σ σ ρσ σ ρσ σ μ μ Σ μ x